Answer:
maximum value , f max = 169 (taking the most out of the 4th power)
minimum value , f min = 169/3 (taking the least out of the 4th power)
Step-by-step explanation:
A) Since both function and restriction are symmetrical with respect to x,y and z, there is no reason for one to be more important than the others and therefore one solution would be x=y=z=λ and thus
x2 + y2 + z2 = 13 → 3λ² = 13 →λ² = 13/3
and f would be
f (x, y, z) = x4 + y4 + z4 = 3λ⁴ = 3*(13/3)²=13²/3=169/3
since x⁴ increases faster than 3*x² , f(x,y,z) would be a minimum
and the maximum value would be obtained taking the most out of x⁴, thus doing 2 coordinates =0 ( can be x=0 and y=0) and
z²= 13
f (x, y, z) = x4 + y4 + z4 = 13² = 169
B) strictly, using Lagrange multipliers
f (x, y, z) = x4 + y4 + z4
g (x, y, z) = x2 + y2 + z2 - 13
F(x,y,z) = f (x, y, z) -λ*g (x, y, z)
such that
Fx (x,y,z)= fx(x, y, z) -λ*gx (x, y, z) = 0 → 4*x³ - λ*2*x = 0 → 2*x*(2*x² -λ) = 0
thus x=0 or x²= λ/2
Fy (x,y,z)= fy(x, y, z) -λ*gy (x, y, z)= 0 → 4*y³ - λ*2*y = 0 → 2*y*(2*y² - λ) = 0
thus y=0 or y²= λ/2
Fz (x,y,z)= fz(x, y, z) -λ*gz (x, y, z)= 0 → 4*z³ - λ*2*z = 0→ 2*z*(2*z² - λ) = 0
thus z=0 or z²= λ/2
g (x, y, z) = 0 → x2 + y2 + z2 = 13 → 3*(λ/2) = 13 → λ=13*2/3
thus x²=y²=z²= λ/2 =13/3
f min = f (x, y, z) = x4 + y4 + z4 = 3*(13/3)²=169/3
for the x=0 , y=0 → z²= 13
f max = f (x, y, z) = x4 + y4 + z4 = 13² = 169
